Mixing
First-Order Stochastic Dominance
Clash Royale CLAN TAG#URR8PPP
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Consider two cumulative distribution functions $F(x)$ and $G(x)$ for $xin[a,b]$ where $G(x)$ has the first-order stochastic dominance over $F(x)$. That is, $F(x)>G(x)$ for all $xin(a,b)$. We assume $a<0$ and $b>0$. Let $f(x)$ and $g(x)$ be the probability density function of $F(x)$ and $G(x)$ respectively.
Suppose the expected value of $x$ under $F(x)$ is positive:
$$
int_a^bxf(x)dx=int_a^0xf(x)dx+int_0^bxf(x)dx>0.
$$
Under this condition, does $f(x)-g(x)>0$ always hold in any interval of $0<x<b$?
Graphical Expression of the Question is Here.
calculus statistics probability-distributions economics
asked Jul 25 at 15:52
Tom M.
163
add a comment |Â
up vote
1
down vote
favorite
Consider two cumulative distribution functions $F(x)$ and $G(x)$ for $xin[a,b]$ where $G(x)$ has the first-order stochastic dominance over $F(x)$. That is, $F(x)>G(x)$ for all $xin(a,b)$. We assume $a<0$ and $b>0$. Let $f(x)$ and $g(x)$ be the probability density function of $F(x)$ and $G(x)$ respectively.
Suppose the expected value of $x$ under $F(x)$ is positive:
$$
int_a^bxf(x)dx=int_a^0xf(x)dx+int_0^bxf(x)dx>0.
$$
Under this condition, does $f(x)-g(x)>0$ always hold in any interval of $0<x<b$?
Graphical Expression of the Question is Here.
calculus statistics probability-distributions economics
asked Jul 25 at 15:52
Tom M.
163
Thank you for your comments. Could you tell me how I can edit my question? Should I delete this question and post a new question?
– Tom M.
Jul 25 at 16:01
1
Please don't delete the question. There is a link that allows you to edit the question just below it. (It is just above and to the left of the box that shows your name.)
– Theoretical Economist
Jul 25 at 16:02
Thank you so much!
– Tom M.
Jul 25 at 16:03
Also, your definition of FOSD seems much stronger than the usual definition. Is that intentional?
– Theoretical Economist
Jul 25 at 16:04
Yes, it is intentional. Thank you for pointing it out.
– Tom M.
Jul 25 at 16:05
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Consider two cumulative distribution functions $F(x)$ and $G(x)$ for $xin[a,b]$ where $G(x)$ has the first-order stochastic dominance over $F(x)$. That is, $F(x)>G(x)$ for all $xin(a,b)$. We assume $a<0$ and $b>0$. Let $f(x)$ and $g(x)$ be the probability density function of $F(x)$ and $G(x)$ respectively.
Suppose the expected value of $x$ under $F(x)$ is positive:
$$
int_a^bxf(x)dx=int_a^0xf(x)dx+int_0^bxf(x)dx>0.
$$
Under this condition, does $f(x)-g(x)>0$ always hold in any interval of $0<x<b$?
Graphical Expression of the Question is Here.
calculus statistics probability-distributions economics
asked Jul 25 at 15:52
Tom M.
163
Consider two cumulative distribution functions $F(x)$ and $G(x)$ for $xin[a,b]$ where $G(x)$ has the first-order stochastic dominance over $F(x)$. That is, $F(x)>G(x)$ for all $xin(a,b)$. We assume $a<0$ and $b>0$. Let $f(x)$ and $g(x)$ be the probability density function of $F(x)$ and $G(x)$ respectively.
Suppose the expected value of $x$ under $F(x)$ is positive:
$$
int_a^bxf(x)dx=int_a^0xf(x)dx+int_0^bxf(x)dx>0.
$$
Under this condition, does $f(x)-g(x)>0$ always hold in any interval of $0<x<b$?
Graphical Expression of the Question is Here.
calculus statistics probability-distributions economics
asked Jul 25 at 15:52
Tom M.
163
edited Jul 25 at 16:09
asked Jul 25 at 15:52
Tom M.
163
asked Jul 25 at 15:52
Tom M.
163
asked Jul 25 at 15:52
Tom M.
163
163
Thank you for your comments. Could you tell me how I can edit my question? Should I delete this question and post a new question?
– Tom M.
Jul 25 at 16:01
1
Please don't delete the question. There is a link that allows you to edit the question just below it. (It is just above and to the left of the box that shows your name.)
– Theoretical Economist
Jul 25 at 16:02
Thank you so much!
– Tom M.
Jul 25 at 16:03
Also, your definition of FOSD seems much stronger than the usual definition. Is that intentional?
– Theoretical Economist
Jul 25 at 16:04
Yes, it is intentional. Thank you for pointing it out.
– Tom M.
Jul 25 at 16:05
add a comment |Â
Thank you for your comments. Could you tell me how I can edit my question? Should I delete this question and post a new question?
– Tom M.
Jul 25 at 16:01
1
Please don't delete the question. There is a link that allows you to edit the question just below it. (It is just above and to the left of the box that shows your name.)
– Theoretical Economist
Jul 25 at 16:02
Thank you so much!
– Tom M.
Jul 25 at 16:03
Also, your definition of FOSD seems much stronger than the usual definition. Is that intentional?
– Theoretical Economist
Jul 25 at 16:04
Yes, it is intentional. Thank you for pointing it out.
– Tom M.
Jul 25 at 16:05
Thank you for your comments. Could you tell me how I can edit my question? Should I delete this question and post a new question?
– Tom M.
Jul 25 at 16:01
Thank you for your comments. Could you tell me how I can edit my question? Should I delete this question and post a new question?
– Tom M.
Jul 25 at 16:01
1
1
Please don't delete the question. There is a link that allows you to edit the question just below it. (It is just above and to the left of the box that shows your name.)
– Theoretical Economist
Jul 25 at 16:02
Please don't delete the question. There is a link that allows you to edit the question just below it. (It is just above and to the left of the box that shows your name.)
– Theoretical Economist
Jul 25 at 16:02
Thank you so much!
– Tom M.
Jul 25 at 16:03
Thank you so much!
– Tom M.
Jul 25 at 16:03
Also, your definition of FOSD seems much stronger than the usual definition. Is that intentional?
– Theoretical Economist
Jul 25 at 16:04
Also, your definition of FOSD seems much stronger than the usual definition. Is that intentional?
– Theoretical Economist
Jul 25 at 16:04
Yes, it is intentional. Thank you for pointing it out.
– Tom M.
Jul 25 at 16:05
Yes, it is intentional. Thank you for pointing it out.
– Tom M.
Jul 25 at 16:05
add a comment |Â
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Thank you for your comments. Could you tell me how I can edit my question? Should I delete this question and post a new question?
– Tom M.
Jul 25 at 16:01
1
Please don't delete the question. There is a link that allows you to edit the question just below it. (It is just above and to the left of the box that shows your name.)
– Theoretical Economist
Jul 25 at 16:02
Thank you so much!
– Tom M.
Jul 25 at 16:03
Also, your definition of FOSD seems much stronger than the usual definition. Is that intentional?
– Theoretical Economist
Jul 25 at 16:04
Yes, it is intentional. Thank you for pointing it out.
– Tom M.
Jul 25 at 16:05